2a+1=a(7+3a)

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Solution for 2a+1=a(7+3a) equation:


Simplifying
2a + 1 = a(7 + 3a)

Reorder the terms:
1 + 2a = a(7 + 3a)
1 + 2a = (7 * a + 3a * a)
1 + 2a = (7a + 3a2)

Solving
1 + 2a = 7a + 3a2

Solving for variable 'a'.

Combine like terms: 2a + -7a = -5a
1 + -5a + -3a2 = 7a + 3a2 + -7a + -3a2

Reorder the terms:
1 + -5a + -3a2 = 7a + -7a + 3a2 + -3a2

Combine like terms: 7a + -7a = 0
1 + -5a + -3a2 = 0 + 3a2 + -3a2
1 + -5a + -3a2 = 3a2 + -3a2

Combine like terms: 3a2 + -3a2 = 0
1 + -5a + -3a2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-0.3333333333 + 1.666666667a + a2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 1.666666667a + 0.3333333333 + a2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 1.666666667a + a2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 1.666666667a + a2 = 0 + 0.3333333333
1.666666667a + a2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
1.666666667a + a2 = 0.3333333333

The a term is 1.666666667a.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667a + 0.6944444447 + a2 = 0.3333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667a + a2 = 0.3333333333 + 0.6944444447

Combine like terms: 0.3333333333 + 0.6944444447 = 1.027777778
0.6944444447 + 1.666666667a + a2 = 1.027777778

Factor a perfect square on the left side:
(a + 0.8333333335)(a + 0.8333333335) = 1.027777778

Calculate the square root of the right side: 1.013793755

Break this problem into two subproblems by setting 
(a + 0.8333333335) equal to 1.013793755 and -1.013793755.

Subproblem 1

a + 0.8333333335 = 1.013793755 Simplifying a + 0.8333333335 = 1.013793755 Reorder the terms: 0.8333333335 + a = 1.013793755 Solving 0.8333333335 + a = 1.013793755 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + a = 1.013793755 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = 1.013793755 + -0.8333333335 a = 1.013793755 + -0.8333333335 Combine like terms: 1.013793755 + -0.8333333335 = 0.1804604215 a = 0.1804604215 Simplifying a = 0.1804604215

Subproblem 2

a + 0.8333333335 = -1.013793755 Simplifying a + 0.8333333335 = -1.013793755 Reorder the terms: 0.8333333335 + a = -1.013793755 Solving 0.8333333335 + a = -1.013793755 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + a = -1.013793755 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = -1.013793755 + -0.8333333335 a = -1.013793755 + -0.8333333335 Combine like terms: -1.013793755 + -0.8333333335 = -1.8471270885 a = -1.8471270885 Simplifying a = -1.8471270885

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.1804604215, -1.8471270885}

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